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XI. PERMEABILITY Soil Mechanics
SEEPAGE AND FLOW NETS
Seepage Terminology Seepage is defined as the flow of a fluid, usually water,
through a soil under a hydraulic gradient.
A hydraulic gradient (i) is supposed to exist between two
points if there exists a difference in the hydraulic head at
the two points.
Hydraulic head is the sum of the position or datum head
and pressure head of water. The discussion on flow nets and seepage relates to the practical aspect of
controlling groundwater during and after construction of foundation below the groundwater table, earth dam
and weirs on permeable foundations.
XI.1. PERMEABILITY Soil Mechanics
SEEPAGE AND FLOW NETS One-dimensional flow
Flow Net for One-Dimensional Flow
Figure 6.1(a) shows a tube of square cross-section (400 mm × 400 mm)
through which steady state vertical flow is occurring. The total head, elevation head and pressure are plotted in Fig. 6.1(b). The rate of
seepage through the tube may be computed by Darcy’s law:
as the situation is one of simple one-dimensional flow.
Flow Net for One-Dimensional Flow
Flow Net for One-Dimensional Flow
In the figure, dashed lines indicate the lines along which the total head is a
constant. These lines through points of equal total head are known as
‘equipotential lines’. Just as the number of flow lines is infinite, the number of
equipotential lines is also infinite.
If equipotential lines are drawn at equal intervals, it means that the head loss
between any two consecutive equipotential lines is the same.
A system of flow lines and equipotential lines, as shown in Fig. 6.1 (c),
constitutes a ‘flow net’ . In isotropic soil, the flow lines and equipotential lines
intersect at right angles, indicating that the direction of flow is perpendicular
to the equipotential lines. An orthogonal net is formed by the intersecting
flow lines and equipotential lines. The simplest of such patterns is one of the
squares. From a flow net three very useful items of information may be
obtained: rate of flow or discharge; head; and hydraulic gradient.
Flow Net for One-Dimensional Flow
First, let us see how to determine the rate of flow or discharge from the flow net. Consider square a
in the flow net – Fig. 6.1(c). The discharge 𝑞𝑎 through this square is,
𝑞𝑎 = 𝑘. 𝑖𝑎. 𝐴𝑎
The head lost in square a is given as 𝐻/𝑁𝑑, where 𝐻 is the total head lost and 𝑁𝑑 is the number of
head drops in the flow net. 𝑖𝑎 is then equal to 𝐻/(𝑁𝑑 . 𝑙), where 𝑙 is the vertical dimension of square 𝑎. The cross-sectional area 𝐴𝑎 of square 𝑎, as seen in plan, is 𝑏 as shown in the figure, since a unit
dimension perpendicular to the plane of the paper is to be considered for the sake of convenience. Thus,
𝑞𝑎 = 𝑘.𝐻
𝑁𝑑 . 𝑙. 𝑏
Since a square net is chosen, 𝑏 = 𝑙. Thus,
𝑞𝑎 = 𝑘.𝐻
𝑁𝑑
If the number of flow channels in a flow net is equal to 𝑁𝑓, the total rate of flow through all the
channels per unit length can be given by,
𝒒 = 𝒌.𝑯.𝑵𝒇
𝑵𝒅
XI.2. PERMEABILITY Soil Mechanics
SEEPAGE AND FLOW NETS Two-dimensional flow
Seepage Terminology
concrete dam
impervious strata
soil
Stream line is simply the path of a water molecule.
datum
H
h = 0 h = H
From upstream to downstream, total head steadily decreases along the stream line.
Equipotential line is simply a contour of constant total head.
concrete dam
impervious strata
soil
datum
H
h = 0 h = H
h=0.8H
Seepage Terminology
Flow Net A network of selected stream lines and equipotential lines.
concrete dam
impervious strata
soil
curvilinear
square
90º
Quantity of Seepage (q)
d
f
N
NHkq .. …. per unit length normal to the plane
# of flow channels
# of equipotential drops
impervious strata
concrete dam
H
head loss from upstream to
downstream
Heads at a Point ‘X’
impervious strata
concrete dam datum
X
z
H
h = H h = 0
Total head, h = H - # of drops from upstream x h
h
Elevation head, he = - z
Pressure head, hp = Total head – Elevation head dN
H
Flow Net for Two-Dimensional Flow
Flow under Sheet Pile Wall
Flow Net for Two-Dimensional Flow
Flow under Concrete Dam
Flow Net for Two-Dimensional Flow
Flow under Concrete Dam
Flow Net for Two-Dimensional Flow
Flow through Earth Dam
Flow Net for Two-Dimensional Flow
Flow through Earth Dam
Flow Net for Two-Dimensional Flow
Flow through Earth Dam
Basic Equation for Seepage
Laplace’s Equation of Continuity
The flow net was introduced in an intuitive manner in the preceding sections. The
equation for seepage through soil which forms the theoretical basis for the flow net as
well as other methods of solving flow problems will be derived in this section.
The following assumptions are made:
1. Darcy’s law is valid for flow through soil.
2. The hydraulic boundary conditions are known at entry and exit of the fluid (water) into the porous
medium (soil).
3. Water is incompressible.
4. The porous medium is incompressible.
These assumptions have been known to be very nearly or precisely valid.
Basic Equation for Seepage
Laplace’s Equation of Continuity
Basic Equation for Seepage
Laplace’s Equation of Continuity
For flow at a point A, we consider an elemental soil block. The block has dimensions dx,
dy, and dz (length dy is perpendicular to the plane of the paper). Let vx and vz be the
components of the discharge velocity in the horizontal and vertical directions,
respectively. The rate of flow of water into the elemental block in the horizontal direction is equal to vx.dz.dy, and in the vertical direction it is vz.dx.dy. The rates of outflow
from the block in the horizontal and vertical directions are:
Assuming that water is incompressible and that no volume change in the soil mass
occurs, we know that the total rate of inflow should equal the total rate of outflow.
Thus,
Basic Equation for Seepage
Laplace’s Equation of Continuity
With Darcy’s law, the discharge velocities can be expressed as:
where 𝑘𝑥 and 𝑘𝑧 are the hydraulic conductivities in the vertical and horizontal
directions, respectively.
If the soil is isotropic with respect to the hydraulic conductivity – that is, 𝑘𝑥 = 𝑘𝑧 - the
continuity equation for two-dimensional flow simplifies to.
𝒒 = 𝒌.𝑯.𝑵𝒇
𝑵𝒅
Basic Equation for Seepage
Laplace’s Equation of Continuity
With Darcy’s law, the discharge velocities can be expressed as:
where 𝑘𝑥 and 𝑘𝑧 are the hydraulic conductivities in the vertical and horizontal
directions, respectively.
For anisotropic soils, 𝑘𝑥 ≠ 𝑘𝑧. In this case, the equation represents two families of curves
that do not meet at 90°. However, we can rewrite the equation as:
𝒒 = 𝒌𝒙. 𝒌𝒛. 𝑯.𝑵𝒇
𝑵𝒅
XI.3. PERMEABILITY Soil Mechanics
SEEPAGE AND FLOW NETS Safety of Hydraulic Structures
Against Piping
Piping in Granular Soils
datum concrete dam
impervious strata
soil
H
At the downstream, near the dam,
h = the head loss between the last
two equipotential lines
l
l
hiexit
the exit hydraulic gradient,
l = the length of the flow element
Piping in Granular Soils
datum
concrete dam
impervious strata
soil
H
If iexit exceeds the critical hydraulic gradient (icr), firstly, the
soil grains at exit get washed away.
no soil; all water
This phenomenon progresses, forming a free passage of water (pipe). If effective stress equal zero, soil stability is
lost leading to ‘boiling’ or ‘quick condition’.
w
cri
'
Piping in Granular Soils Piping is a very serious problem. It leads to downstream
flooding which can result in loss of lives.
concrete dam
impervious strata
soil
Therefore, provide adequate safety factor against piping.
exit
crpiping
i
iSF ..
typically 3~4
Stresses due to Flow
X
soil
hw
L
Static Situation (No flow)
z v = whw + satz
u = w (hw + z)
v ' = ' z
At X,
Introduction Introduction
Downward Flow
hw
L
flow
X
soil
z
v = whw + satz
w hw + w(L-H)(z/L)
v ' = ' z + wiz
At X,
H u = w hw
u = w (hw+L-H)
… as for static case
= w hw + w(z-iz)
= w (hw+z) - wiz Reduction due to flow
Increase due to flow
u =
Stresses due to Flow Introduction Introduction
flow Upward Flow
hw
L X
soil
z
v = whw + satz
= w hw + w(L+H)(z/L)
v ' = ' z - wiz
At X, H
u = w hw
u = w (hw+L+H)
… as for static case
= w hw + w(z+iz)
= w (hw+z) + wiz
Increase due to flow
Reduction due to flow
u
Stresses due to Flow Introduction Introduction
Quick Condition in Granular Soils During upward flow, at X:
v ' = ' z - wiz
flow
hw
L X
soil
z
H
izw
w
'
Critical hydraulic gradient (icr)
If i > icr, the effective stress is negative.
i.e., no intergranular contact, thus, failure.
- Quick Condition -
Problem 1 Draw the total, elevation and pressure heads
diagrams for the figure shown. Also,
determine the following:
1.1 Hydraulic gradient (2 m/m)
1.2 Discharge velocity (1 cm/sec)
1.3 Seepage velocity (3 cm/sec)
Problem Set 8
1.2 m
1.8 m
0.6 m
Problem 2 Draw the total, elevation and pressure heads
diagrams for the figure shown.
Problem Set 8
1.2 m
1.8 m
0.6 m
1.2 m
Problem 3 A flow net for flow around a single row
of sheet piles in a permeable soil
layer is shown in the figure. We are given that kx = kz = k = 5x10-3 cm/sec.
Determine:
3.1 How high (above the ground surface) will
the water rise if piezometers are placed at
points a, b, c, and d?
3.2 What is the rate of seepage through flow
channel II per unit length (perpendicular to the
section shown)?
Problem Set 8
Problem 4 A flow net for flow around a
single row of sheet piles in a
permeable soil layer is shown
in the figure. We are given that kx = kz = k = 5x10-9 m/sec.
Determine:
4.1 The pressure at a ~ l.
4.2 The total flow rate.
4.3 The hydraulic gradient at ‘r’
element.
Problem Set 8
El. 0 m
El. 11.7 m
El. 14.7 m
El. 18 m
El. 27 m
El. 19.5 m
El. 9 m
c
d f
g
Problem 5 A flow net for flow around a
concrete in a permeable
soil layer is shown in the figure. Determine the uplift
pressure under the concrete
dam.
Problem Set 8
impervious strata
concrete dam
14 m
El. 0 m
El. 10 m
El. 17 m
El. 8 m a b c d e f